Editing G-structure on a manifold (section)

==== In terms of concrete bundles ====
Given a principal <math>G</math>-bundle <math>P</math> over <math>X</math>, a ''reduction of the structure group'' (from <math>G</math> to <math>H</math>) is a ''<math>H</math>''-bundle <math>Q</math> and an isomorphism <math>\phi_Q\colon Q \times_H G \to P</math> of the [[associated bundle]] to the original bundle.